THE BRAIDINGS IN THE MAPPING CLASS GROUPS OF SURFACES

Title & Authors
THE BRAIDINGS IN THE MAPPING CLASS GROUPS OF SURFACES
Song, Yongjin;

Abstract
The disjoint union of mapping class groups of surfaces forms a braided monoidal category $\small{\mathcal{M}}$, as the disjoint union of the braid groups $\small{\mathcal{B}}$ does. We give a concrete and geometric meaning of the braidings $\small{{\beta}_{r,s}}$ in $\small{\mathcal{M}}$. Moreover, we find a set of elements in the mapping class groups which correspond to the standard generators of the braid groups. Using this, we can define an obvious map $\small{{\phi}\;:\;B_g{\rightarrow}{\Gamma}_{g,1}}$. We show that this map $\small{{\phi}}$ is injective and nongeometric in the sense of Wajnryb. Since this map extends to a braided monoidal functor $\small{{\Phi}\;:\;\mathcal{B}{\rightarrow}\mathcal{M}}$, the integral homology homomorphism induced by $\small{{\phi}}$ is trivial in the stable range.
Keywords
braid group;mapping class group;Dehn twists;braided monoidal category;double loop space;plus construction;
Language
English
Cited by
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